# Isometric and affine copies of a set in volumetric Helly results

@article{Messina2022IsometricAA,
title={Isometric and affine copies of a set in volumetric Helly results},
author={John A. Messina and Pablo Sober'on},
journal={Comput. Geom.},
year={2022},
volume={103},
pages={101855}
}
• Published 8 October 2020
• Mathematics
• Comput. Geom.

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A version of the “quantitative” diameter theorem of Bárány, Katchalski and Pach, with a polynomial dependence on the dimension.

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We prove the following Helly-type result. Let $\mathcal{C}_1,\ldots,\mathcal{C}_{3d}$ be finite families of convex bodies in $\mathbb{R}^d$. Assume that for any colorful choice of $2d$ sets,

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We provide a number of new quantitative versions of Helly's theorem. For example, we show that for every family $\{P_i:i\in I\}$ of closed half-spaces P_i=\{x\in {\mathbb R}^n:\langle x,w_i\rangle

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INTRODUCTION Let F be a family of convex sets in R. A geometric transversal is an affine subspace that intersects every member of F . More specifically, for a given integer 0 ≤ k < d, a k-dimensional

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Bárány, Katchalski and Pach (Proc Am Math Soc 86(1):109–114, 1982) (see also Bárány et al., Am Math Mon 91(6):362–365, 1984) proved the following quantitative form of Helly’s theorem. If the

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We discuss a problem posed by Bárány, Katchalski and Pach: if {Pi : i ∈ I} is a family of closed convex sets in R such that diam (⋂ i∈I Pi ) = 1 then there exist s 6 2n and i1, . . . , is ∈ I such

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• 2017
We prove variations of Carathéodory’s, Helly’s and Tverberg’s theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we